An electron tunneling across a junction integrated into an electric circuit can generate an excitation in the photonic field (electromagnetic environment) and lose energy in the process. Such inelastic tunneling of particles is commonly described using the P(E) theory. In the conventional approach to this theory, the tunneling rate and the electric current through the junction are derived using Fermi's golden rule and by averaging over the environmental photonic degrees of freedom. In this work, we address the same problem of inelastic tunneling due to photonic environment in Lindbladian formalism and we present how the photonic degrees of freedom are traced out in the quantum master equation approach. The resulting quantum master equation is parametrized by the same P(E) function and enables us to obtain not only the electric current but various other quantities, for instance, the heat current, in a systematic and convenient way. We also demonstrate that the Lindbladian formalism provides a comprehensive description of Bogoliubov quasiparticle tunneling through superconducting junctions and that it properly accounts for the coherence factors. The coherence factors become important if the normal-state density of states is particle-hole asymmetric.

We present a theoretical study of a system consisting of a superconducting island and two quantum dots, a possible platform for building qubits and Cooper pair splitters, in the regime where each dot hosts a single electron and, hence, carries a magnetic moment. We focus on the case where the dots are coupled to overlapping superconductor states and we study whether the spins are ferromagnetically or antiferromagnetically aligned. We show that if the total number of particles is even the spins align antiferromagnetically in the flatband limit, i.e., when the bandwidth of the superconductor is negligibly small, but they align ferromagnetically if the bandwidth is finite and above some value. If the total number of particles is odd, the alignment is ferromagnetic independently from the bandwidth. This implies that the results of the flatband limit are applicable only within a restricted parameter regime for realistic superconducting qubit systems and that some care is required in applying simplified models to devices such as Cooper pair splitters.

We use the Schrieffer-Wolff transformation (SWT) to analyze Josephson junctions between superconducting leads described by the charge-conserving BCS theory. Starting from the single-electron tunneling terms, we directly recover the conventional effective Hamiltonian, −EJ cos ϕ̂, with an operator-valued phase bias ϕ̂. The SWT approach has the advantage that it can be systematically extended to more complex scenarios. We show that if a Bogoliubov quasiparticle is present its motion couples to that of Cooper pairs, introducing correlated dynamics that reshape the energy spectrum of the junction. Furthermore, higher-order terms in the SWT naturally describe Josephson harmonics, whose amplitudes are directly related to the microscopic properties of the superconducting leads and the junction. We derive expressions that could facilitate tuning the ratio between the different harmonics in a controlled way.